Question

A new LED light to replace incandescent bulbs has come on the market. The box says it has an average life of 9,000 hours with a standard deviation of 300 hours.

a) What is the probability that a single bulb will last between 8950 and 9100 hours?

b) What is the probability that the mean of a sample of 70 bulbs picked at random will be between 8950 and 9100 hours?

Answer 1

Given, mean = 9000 , sd=300

a) To find probability that a single bulb will last between 8950 and 9100 hours?

Let X: Life of LED bulb. Then assuming X has Normal distribution with mean 9000 and standard deviation 300.

P(8950 < x < 9100 )= P(x<9100) - P(x < 8950 )

Using Excel formula,

General formula for calculating probability is,

P(X<x) = NORMDIST (x, mean , standard deviation, Cumulative=1)

P( x< 9100 ) = NORMDIST (9100,9000,300,1)

P( x < 9100) = 0.6305

P( x< 8950 ) = NORMDIST( 8950,9000,300,1)

P( x < 8950) = 0.4338

P(8950 <x< 9100 ) = 0.1967

b) Here we have to find Probability of sample mean based on sample of size 70, so sampling distribution of X bar is ,

sd(x bar ) =300/√70 = 35.85

Now we have to find probability of sample mean lies between 8950 & 9100.

i.e. P(8950< x bar < 9100) =

P( x bar < 9100) - P( x bar <8950 )

P( xbar < 9100 ) = 0.9973

NORMDIST (9100,9000,35.85,1)= 0.9973

P( xbar < 9100) = 0.9973

P( xbar< 8950 ) = NORMDIST( 8950,9000,35.85,1) =0.08159

P( xbar < 8950) = 0.08159

P(8950 <xbar< 9100 ) = 0.9157 .